A near-ring $N$ is called strongly reduced if, for $ain N$, $a^2in N_c$ implies $ain N_c$, where $N_c$ denotes the constant part of $N$. We investigate some properties of strongly reduced near-rings and apply those to the study of left strongly regular near-rings. Finally we classify all reduced, and strongly reduced near-rings of order $le 7.$

Keywords: reduced, strongly reduced, left strongly regular, strongly regular